2. “If…then….” and “It is not the case….” 2.1 The Conditional As we noted in chapter 1, there are sentences of a natural language like English
that are not atomic sentences. Our examples included
If Lincoln wins the election, then Lincoln will be President.
The Earth is not the center of the universe.
We could treat these like atomic sentences, but then we would lose a great deal of
important information. For example, the first sentence tells us something about the
relationship between the atomic sentences “Lincoln wins the election” and “Lincoln will
be President.” And the second sentence above will, one supposes, have an interesting
relationship to the sentence, “The Earth is the center of the universe”. To make these
relations explicit, we will have to understand what “if…then…” and “not” mean. Thus, it
would be useful if our logical language was able to express these kinds of sentences also,
in a way that made these elements explicit. Let us start with the first one.
The sentence “If Lincoln wins the election, then Lincoln will be President”
contains two atomic sentences, “Lincoln wins the election” and “Lincoln will be
President.” We could thus represent this sentence by letting
Lincoln wins the election
be represented in our logical language by
S
And by letting
Lincoln will be president
be represented by
T
Then, the whole expression could be represented by writing
If S then T
It will be useful, however, to replace the English phrase “if…then...” by a single symbol
in our language. The most commonly used such symbol is “→”. Thus, we would write
S→T
One last thing needs to be observed, however. We do not yet know whether the order of
the sentences in this expression matters. We need to be clear then that this “→” symbol
is meant to relate only these two atomic sentences. There are several ways to do this, but
the most familiar (although not the most elegant) is to use parentheses. Thus, we will
write our expression
(S→T)
This kind of sentence is called a “conditional”. The first constituent sentence (the
one before the arrow, which in this example is “S”) is called the “antecedent”. The
second sentence (the one after the arrow, which in this example is “T”) is called the
“consequent”.
We know how to write the conditional, but what does it mean? As before, we will
take the meaning to be given by the truth conditions—that is, a description of when the
sentence is either true or false. We do this with a truth table. But now, our sentence has
two parts, S and T. Note that either sentence could be true or false. That means, we have
to consider four possible kinds of situations. We must consider when T is true and when
it is false, but then we need to consider those two kinds of situations twice: once for
when S is true and once for when S is false. Thus, the left hand side of our truth table
will look like this:
S
T
T
F
F
T
T
F
T
F
There are four kinds of ways the world could be that we must consider.
Note that, since there are two possible true values, whenever we consider another
atomic sentence, there are twice as many ways the world could be that we should
consider. Thus, for n atomic sentences, our truth table must have 2n rows.
Now, we must decide upon what the conditional means. To some degree this is
up to us. What matters is that once we define the semantics of the conditional, we stick
to our definition. But we want to capture as much of the meaning of the English
“if…then…” as we can, while remaining absolutely precise in our language.
Let us consider each kind of way the world could be. For the first row of the truth
table, we have that S is true and T is true. Suppose the world is such that Lincoln wins
the election, and also Lincoln will be President. Then, would I have spoken truly if I
said, “If Lincoln wins the election, then Lincoln will be President”? Most people agree
that I would have. Similarly, suppose that Lincoln wins the election, but Lincoln will not
be President. Would the sentence “If Lincoln wins the election, then Lincoln will be
President” still be true? Most agree that it would be false now. So the first rows of our
truth table are uncontroversial.
S
T
T
F
F
T
T
F
T
F
(S→T)
T
F
Trouble begins with the next two rows. Note now that our principle of bivalence
requires us to fill in these rows. We cannot leave them blank. If we did, we would be
saying that sometimes a conditional can have no truth value. So, if we are going to
respect the principle of bivalence, then we have to put either T or F in for each of the last
two rows.
It is helpful at this point to change our example. Consider another sentence,
which we will suppose is about numbers. In particular, assume that a is a particular
natural number, only you and I don’t know what number it is. Consider now the
following sentence.
If a is evenly divisible by 4, then a is evenly divisible by 2.
(By “evenly divisible,” I mean divisible without remainder.) The first thing to ask
yourself is: is this sentence true? I hope we can all agree that it is—even though we do
not know what a is. Let
a is evenly divisible by 4
be represented in our logic by
U
and let
a is evenly divisible by 2
be represented by
V
Our sentence then is
(U→V)
And its truth table—as far as we understand right now—is:
U
T
T
V
T
F
(U→V)
T
F
F
F
T
F
Now consider a case in which a is 6. This is like the third row of the truth table. It is not
the case that 6 is evenly divisible by 4, but it is the case that 6 is evenly divisible by 2.
And consider the case in which a is 7. This is like the fourth row of the truth table; 7
would be evenly divisible by neither 4 nor by 2. But we agreed that the conditional is
true—regardless of the value of a! So, the truth table must be
U
T
T
F
F
V
T
F
T
F
(U→V)
T
F
T
T
(One thing is a little funny about this example. We agreed on the meaning of the second
row of the table for any conditional before we considered this specific example.
However, we will not be able to find a number such that it is evenly divisible by 4 and
not evenly divisible by 2, so the world will never be like the second row describes. Two
things need to be said about that. First, this oddity arises because of mathematical facts,
not purely logical ones—that is, we need to know what “divisible” means, what “4” and
“2” mean, and so on, in order to understand the sentence. So, when we see that the
second row is not possible, we are basing that on our knowledge of mathematics, not on
our knowledge of propositional logic. Second, some conditionals can be false, and
therefore we need to consider all possible cases, not just special conditionals like this
example from arithmetic. So, we must define the conditional for any case where the
antecedent is true and the consequent is false, even if that cannot happen for this specific
example.)
Following this pattern, we should also fill out our table about the election with
S
T
T
F
F
T
T
F
T
F
(S→T)
T
F
T
T
If you are dissatisfied by this, it might be helpful to think of these last two rows as
vacuous cases. A conditional tells us about what happens if the antecedent is true. But
when the antecedent is false, we simply default to true.
We are now ready to offer, in a more formal way, the syntax and semantics for the
conditional.
The syntax of the conditional is that, if Φ and Ψ are sentences, then
(Φ→Ψ)
is a sentence.
The semantics of the conditional are given by a truth table. For any sentences Φ
and Ψ:
Φ
T
T
F
F
Ψ
T
F
T
F
(Φ→Ψ)
T
F
T
T
Remember that this truth table is now a definition. We are agreeing to use the symbol
“→” to mean this from here on out.
The elements of the propositional logic, like “→”, that we add to our language in
order to form more complex sentences, are called “truth functional connectives”. I hope
it is clear why: the meaning of this symbol is given in a truth function. (If you are
unfamiliar or uncertain about the idea of a function, think of a function as like a machine
that takes in one or more inputs, and always then gives exactly one output. For the
conditional, the inputs are two truth values; and the output is one truth value. For
example, put T F into the truth function called “→”, and you get out F.)
2.2 Alternative phrasings in English for the conditional. Only if. English includes many alternative phrases that appear to be equivalent to the
conditional. Furthermore, in English and other natural languages, the order of the
conditional will sometimes be reversed. We can capture the general sense of these cases
by recognizing that each of the following phrases would be translated as (P→Q).
If P, then Q.
Q, if P.
On the condition that P, Q.
Q, on the condition that P.
Given that P, Q.
Q, given that P.
Provided that P, Q.
Q, provided that P.
When P, then Q.
Q, when P.
P implies Q.
Q is implied by P.
P is sufficient for Q.
Q is necessary for P.
Another oddity of English is that the word “only” changes the meaning of “if”.
You can see this if you consider the following two sentences.
Fifi is a cat, if Fifi is a mammal.
Fifi is a cat only if Fifi is a mammal.
Suppose we know Fifi is an organism, but don’t know what kind of organism Fifi is. It
could be a dog, a cat, a gray whale, a ladybug, a sponge. It seems clear that the first
sentence is not necessarily true. If Fifi is a gray whale, for example, then it is true that
Fifi is a mammal, but false that Fifi is a cat; and so, the first sentence would be false. But
the second sentence looks like it must be true.
We should thus be careful to recognize that “only if” does not mean the same
thing as “if”. (If it did, these two sentences would have the same truth value in all
situations.) In fact, it seems that “only if” can best be expressed by a conditional where
the “only if” appears before the consequent (remember, the consequent is the second part
of the conditional—the part that the arrows points at). Thus, sentences of this form:
P only if Q.
Only if Q, P.
are best expressed by the formula
(P→Q)
2.3 Test your understanding of the conditional People tend to find conditionals confusing. In part this seems to be because we
easily confuse them with another kind of relation, which we will learn about later, called
the “biconditional”. Also, sometimes “if…then…” is used in English in different ways
(see section 23.1 if you are curious about alternative possible meanings). But from now
on, we will understand the conditional as described above. To test whether you have
properly grasped the conditional, consider the following puzzle.
We have a set of four cards. Each card has the following property: it has a shape
on one side, and a letter on the other side. We shuffle and mix the cards, flipping some
over while we shuffle. Then, we lay out the four cards:
[Illustration 1 here]
Given our constraint that each card has a letter on one side and a shape on the other, we
know that card 1 has a shape on the unseen side; card 2 has a letter on the unseen side;
and so on.
Consider now the following claim:
For each of these four cards, if the card has a Q on the letter side of the card, then
it has a square on the shape side of the card.
Here is our puzzle: what is the minimum number of cards that we must turn over
to test whether this claim is true, and which cards are they? Of course we could turn
them all over, but the puzzle asks you to identify all and only the cards that will test the
claim.
Stop reading now, and see if you can decide on the answer. Be warned, people
generally perform very poorly on this puzzle. Think about it for a while. The answer is
given below in problem 1.
2.4 Negation In chapter 1, we considered as an example the sentence
The Earth is not the center of the universe.
At first glance, such a sentence might appear to be fundamentally unlike a conditional. It
does not contain two sentences, but only one. There is a “not” in the sentence, but it is
not connecting two sentences. However, we can still think of this sentence as being
constructed with a truth functional connective, if we are willing to accept that this
sentence is equivalent to the following sentence.
It is not the case that the Earth is the center of the universe.
If this sentence is equivalent to the one above, then we can treat “It is not the
case” as a truth functional connective. It is traditional to replace this cumbersome
English phrase with a single symbol, “¬”. Then, mixing our propositional logic with
English, we would have
¬The Earth is the center of the universe.
And if we let W be a sentence in our language that has the meaning “The Earth is the
center of the universe”, we would write
¬W
This connective is called “negation”. Its syntax is: if Φ is a sentence, then
¬Φ
is a sentence. Its semantics is also obvious. If Φ is a sentence, then
Φ
T
F
¬Φ
F
T
To deny a true sentence is to speak a falsehood. To deny a false sentence is to say
something true.
Our syntax always has a property that is called “recursive”. This means that the
rule can be applied repeatedly, to the product of the rule. In other words, our syntax tells
us that if P is a sentence, then ¬P is a sentence. But now note that the same rule applies
again: if ¬P is a sentence, then ¬¬P is a sentence. And so on. Similarly, if P and Q are
sentences, the syntax for the conditional tells us that (P→Q) is a sentence. But then so is
¬(P→Q), and so is (¬(P→Q) → (P→Q)). And so on. If we have just a single atomic
sentence, our recursive syntax will allow us to form infinitely many different sentences
with negation and the conditional.
2.5 Problems 1. The answer to our card game was: you need only turn over cards 3 and 4. This
might seem confusing to many people at first. But remember the meaning of the
conditional: it can only be false if the first part is true and the second part is false.
The sentence we want to test is “For each of these four cards, if the card has a Q
on the letter side of the card, then it has a square on the shape side of the card.”
Let Q stand for “the card has a Q on the letter side of the card.” Let S stand for
“the card has a square on the shape side of the card.” Then we could make a truth
table to express the meaning of the claim being tested:
Q
T
T
F
F
S
T
F
T
F
(Q→S)
T
F
T
T
Look back at the card. The first card has an R on the letter side. So, sentence Q
is false. But then we are looking at the last two rows of the truth table, and the
conditional cannot be false. We do not need to check that card. The second card
has a square on it. That means S is true for that card. But then we are in a
situation represented by either the first or third row of the truth table. Again, the
claim that (Q→S) cannot be false in either case, so there is no point in checking
that card. The third card shows a Q. It corresponds to a situation that is like
either the first or second row of the truth table. We cannot tell then whether
(Q→S) is true or false without turning the card over. Similarly, the last card
shows a situation where S is false, so we are in a kind of situation represented by
either the second or last row of the truth table. We must turn the card over to
determine if (Q→S) is true or false.
Try this puzzle again. Consider the following claim about those four
cards: If there is a star on the shape side of the card, then there is an R on the
letter side of the card. How many cards must you turn over to check this claim?
What cards are they?
2. Consider the following four cards. Each card has a letter on one side, and a shape
on the other side.
[Illustration 2 here]
For each of the following claims, determine (1) How many cards you must turn
over to check the claim, and (2) what those cards are, in order to determine if the
claim is true of all four cards.
a. There is not a Q on the letter side of the card.
b. There is not an octagon on the shape side of the card.
c. If there is a triangle on the shape side of the card, then there is a P on the
letter side of the card.
d. There is an R on the letter side of the card only if there is a diamond on the
shape side of the card.
e. There is a hexagon on the shape side of the card, on the condition that
there is a P on the letter side of the card.
f. There is a diamond on shape side of the card only if there is a P on the
letter side of the card.
3. Which of the following have correct syntax? Which have incorrect syntax?
a. P→Q
b. ¬(P→Q)
c. (¬P→Q)
d. (P¬→Q)
e. (P→¬Q)
4. Use the following translation key to translate the following sentences into a
propositional logic.
Logic
P
Q
a.
b.
c.
d.
e.
Translation Key
English
Abe is able.
Abe is honest.
Abe is honest only if Abe is able.
Abe is not able.
Abe is not able only if Abe is not honest.
Abe is able, provided that Abe is not honest.
If Abe is not able then Abe is not honest.
5. Make up your own translation key to translate the following sentences into a
propositional logic.
a. Josie is a cat.
b. Josie is a mammal.
c. Josie is not a mammal.
d. If Josie is not a cat, then Josie is not a mammal.
e. Josie is a fish.
f. Provided that Josie is a mammal, then Josie is not a fish.
g. Josie is a cat only if Josie is a mammal.
h. Josie is a fish only if Josie is not a mammal.
i. It’s not the case that Jose is not a mammal.
j. Josie is not a cat, if Josie is a fish.
6. Make up your own translation key in order to translate the following sentences
into English. Write out the English equivalents in English sentences that seem (as
much as is possible) natural.
a. (R→S)
b. ¬¬R
c. (S→R)
d. (¬S→¬R)
e. ¬(R→S)